# Po-group

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partially ordered group

A group endowed with a partial order such that for all , (Cf. also Partially ordered group.) If is the identity of a -group and is the positive cone of (cf. -group), then the following relations hold:

1) ;

2) ;

3) for all .

If, in a group , one can find a set with the properties 1)–3), then can be made into a -group by setting if and only if . It is correct to identify the order of a -group with its positive cone. One often writes a -group with positive cone as .

A mapping from a -group into a -group is an order homomorphism if is a homomorphism of the group and for all , A homomorphism from a -group into a -group is an order homomorphism if and only if .

A subgroup of a -group is called convex (cf. Convex subgroup) if for all with , If is a convex subgroup of a -group , then the set of right cosets of by is a partially ordered set with the induced order if there exists an such that . The quotient group of a -group by a convex normal subgroup is a -group respect with the induced partial order, and the natural homomorphism is an order homomorphism. The homomorphism theorem holds for -groups: if is an order homomorphism from a -group into a -group , then the kernel of is a convex normal subgroup of and there exists an order isomorphism from the -group into such that .

The most important classes of -groups are the class of lattice-ordered groups (cf. -group) and the class of totally ordered groups (cf. -group).